If the directed line makes an angle 45^ and 60^ with the X and Y -axes respectively, then the obtuse angle made by the line with the Z -axis is

Step-by-step solution

Let the angles between the directed line and the X, Y, and Z-axes be , , and respectively, with = 45^ and = 60^ given. The direction cosines satisfy the identity ^2 + ^2 + ^2 = 1 . Substituting the known values: ^2 45^ + ^2 60^ + ^2 = 1 . Since 45^ = 1 2 and 60^ = 1 2 , this becomes: ( 1 2 )^2 + ( 1 2 )^2 + ^2 = 1 1 2 + 1 4 + ^2 = 1 3 4 + ^2 = 1 Solving for ^2 yields ^2 = 1 4 , so = 1 2 . The acute angle solution = 60^ corresponds to = 1 2 . The obtuse angle corresponds to = - 1 2 , giving = 120^ . Final answer: 12